Convex fuzzifying bornologies and its duality

The main objective of this paper is to investigate convex fuzzifying bornological linear spaces and their duality. First, we introduce the notions of convex fuzzifying bornological linear spaces and locally convex fuzzifying topological linear spaces, along with several examples. Next, we study the relationship between convex fuzzifying bornological linear spaces and locally convex fuzzifying topological linear spaces, utilizing the mapping of fuzzifying bornivorous. Subsequently, we introduce the concepts of fuzzifying bornological locally convex spaces and fuzzifying topological convex bornological linear spaces. We demonstrate that the category of fuzzifying bornological locally convex spaces can be embedded as a reflective subcategory in the category of convex fuzzifying bornological linear spaces, and the category of fuzzifying topological convex bornological linear spaces can also be embedded as a reflective subcategory in the category of locally convex fuzzifying topological linear spaces. Moreover, the category of fuzzifying topological convex bornological linear spaces is topological over the category of linear spaces with respect to the expected forgetful functor. Lastly, we provide several characterizations of the fuzzifying bornological topologies.